# Variations on the Fundamental Theorem

The importance of the fundamental theorem in single variable calculus cannot be overstated.  It allows us to compute areas, volumes, centroids, arclength, and probability integrals.  It is the basis for theoretical concepts such as improper integrals, Taylor's theorem, and Fourier Series,

However, it may be that the Fundamental theorem is most valuable when it is used in a multivariable setting.  In this chapter, we explore the fundamental theorem in more than one variable by examing its relationship to curves, surfaces, and solids.  This exploration culminates in one of the most important ideas in all of mathematics and science, which is a variation of the fundamental theorem known as Stoke's theorem.  Along the way, we encounter a very important special case of Stoke's theorem known as Green's theorem, and we examine some of its applications.

 5.1 Vector Fields 5.5 Surface Integrals 5.2 Line Integrals 5.6 The Divergence Theorem 5.3 Potentials of Conservative Fields 5.7 Stoke's Theorem 5.4 Green's Theorem 5.8 Differential Forms

Summary and Review